We introduce a one-dimensional (1D) extended quantum breakdown model comprising a fermionic and a spin degree of freedom per site, and featuring a spatially asymmetric breakdown-type interaction between the fermions and spins. Our model resembles the breakdown process of particles incident into a cloud chamber with nonzero quantum amplitudes of both exciting and not exciting the local vapor atoms. We analytically show that, in the absence of any magnetic field for the spins, the model exhibits Hilbert space fragmentation within each symmetry sector into exponentially many Krylov subspaces and hence displays nonthermal dynamics. Here, we demonstrate that the fragmentation naturally occurs in an entangled basis and thus provides an example of ``quantum fragmentation.'' Besides establishing the nature of fragmentation analytically, we also study the long-time behavior of the entanglement entropy and its deviation from the expected Page value as a probe of ergodicity in the system. Upon introducing a magnetic field for the spins, most of the Krylov subspaces merge and the model becomes chaotic. Finally, we study the effects of strong randomness on the system and observe behavior similar to that of many-body localized systems.
We introduce a one-dimensional (1D) extended quantum breakdown model comprising a fermionic and a spin degree of freedom per site, and featuring a spatially asymmetric breakdown-type interaction between the fermions and spins. Our model resembles the breakdown process of particles incident into a cloud chamber with nonzero quantum amplitudes of both exciting and not exciting the local vapor atoms. We analytically show that, in the absence of any magnetic field for the spins, the model exhibits Hilbert space fragmentation within each symmetry sector into exponentially many Krylov subspaces and hence displays nonthermal dynamics. Here, we demonstrate that the fragmentation naturally occurs in an entangled basis and thus provides an example of ``quantum fragmentation.'' Besides establishing the nature of fragmentation analytically, we also study the long-time behavior of the entanglement entropy and its deviation from the expected Page value as a probe of ergodicity in the system. Upon introducing a magnetic field for the spins, most of the Krylov subspaces merge and the model becomes chaotic. Finally, we study the effects of strong randomness on the system and observe behavior similar to that of many-body localized systems.
We propose a two-dimensional (2D) quantum breakdown model of hardcore bosons interacting with disordered spins which would be classical without the bosons. It resembles particles incident into supersaturated vapor. The model exhibits a set of subsystem symmetries and has a strong fragmentation into Krylov subspaces in each symmetry sector. The Hamiltonian in each Krylov subspace maps to a single-particle problem in a Cayley-tree-like graph. At zero disorder, the Krylov subspaces exhibit either (possible) integrable features or quantum chaos with quantum scar states showing irregular energy and degeneracy patterns. At nonzero disorders, they enter a 2D many-body localization (MBL) phase beyond certain disorder strength W∗, as indicated by Poisson level spacing statistics and entanglement entropy growing as logt with time t. Our theoretical arguments suggest W∗ is finite or zero for boson number ${N}_{b}\ensuremath{\lesssim}{L}^{\ensuremath{\gamma}}/logL$ ($1/2\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$) as system size $L\ensuremath{\rightarrow}\ensuremath{\infty}$. This gives a more stringent condition for MBL than that in the 1D quantum breakdown models. This model reveals the possibility of MBL in systems of quantum particles interacting with classical degrees of freedom.
We find that rank deficiency of the local Hamiltonian in a classically fragmented model is the key mechanism leading to quantum Hilbert space fragmentation. The rank deficiency produces local null directions that can generate entangled frozen states (EFS): entangled states embedded in mobile classical Krylov sectors that do not evolve under Hamiltonian dynamics. When the entangled frozen subspace is non-empty, the mobile classical sector splits into a mobile quantum Krylov subspace and an entangled frozen subspace, and the model exhibits quantum fragmentation. We establish this mechanism in four models of increasing symmetry structure: an asymmetric qubit projector with no symmetry, the Z2-symmetric GHZ projector, a Z3-symmetric cyclic qutrit projector, and the Temperley-Lieb model. For the asymmetric and GHZ projector models, we obtain closed-form expressions for irreducible Krylov dimensions, degeneracies, and sector multiplicities. The all-mobile-sector EFS in these two models exhibits a sub-volume-law bipartite entanglement entropy scaling as S∼L. Further, we introduce the notion of weak and strong quantum fragmentation, the quantum counterpart of the weak-strong distinction in classical fragmentation. After removing the EFS, the mobile quantum Krylov subspace decomposes into irreducible blocks. In the weak case, the number of irreducible blocks remains O(1), each is individually ergodic with Gaussian Orthogonal Ensemble (GOE) level statistics, and the unresolved spectrum follows an mGOE distribution. In the strong case, the number of irreducible blocks grows with system size, and the gap-ratio distribution approaches Poisson as L→∞.
We propose a quantum model of fermions simulating the electrical breakdown process of dielectrics. The model consists of M sites with N fermion modes per site and has a conserved charge Q. It has an on-site chemical potential $\ensuremath{\mu}$ with disorder W and an interaction of strength J restricting each fermion to excite two more fermions when moving forward by one site. We show that the N=3 model with disorder W=0 shows a Hilbert space fragmentation and is exactly solvable except for very few Krylov subspaces. The analytical solution shows that the N=3 model exhibits many-body localization (MBL) as $M\ensuremath{\rightarrow}\ensuremath{\infty}$, which is stable against W>0 as our exact diagonalization (ED) shows. At N>3, our ED suggests an MBL to quantum chaos crossover at small W as M/N decreases across 1, and persistent MBL at large W. At W=0, an exactly solvable many-body scar flat band exists in many charge Q sectors, which has a nonzero measure in the thermodynamic limit. We further calculate the time evolution of a fermion added to the particle vacuum, which shows a breakdown (dielectric) phase when $\ensuremath{\mu}/J<1/2$ ($\ensuremath{\mu}/J>1/2$) if $W\ensuremath{\ll}J$, and no breakdown if $W\ensuremath{\gg}J$.
Hilbert space fragmentation refers to exponential growth in the number of dynamically disconnected Krylov sectors with system size. It is taken as evidence of ergodicity breaking, since conventional symmetries generate at most a polynomial number of sectors. However, we demonstrate that generalized symmetries can fragment the Hilbert space. Models with higher-form, subsystem, and gauge symmetries can have exponentially many symmetry sectors. We further prove that non-invertible symmetries can induce additional fragmentation within individual symmetry sectors. Fragmentation in several known models arises from generalized symmetries, and the presence of exponentially many Krylov sectors therefore does not by itself imply ergodicity breaking. Finally, we show that disorder free localization arises naturally from Krylov-restricted thermalization when sectors lack translation invariance, requiring neither ergodicity breaking nor gauge symmetry.
Abstract The one-dimensional quantum breakdown model, which features spatially asymmetric fermionic interactions simulating the electrical breakdown phenomenon, exhibits an exponential U(1) symmetry and a variety of dynamical phases including many-body localization and quantum chaos with quantum scar states. We investigate the minimal quantum breakdown model with the minimal number of on-site fermion orbitals required for the interaction and identify a large number of local conserved charges in the model. We then reveal a mapping between the minimal quantum breakdown model in certain charge sectors and a quantum link model which simulates the U(1) lattice gauge theory and show that the local conserved charges map to the gauge symmetry generators. A special charge sector of the model further maps to the PXP model, which shows quantum many-body scars. This mapping unveils the rich dynamics in different Krylov subspaces characterized by different gauge configurations in the quantum breakdown model.
We show that the combination of charge and dipole conservation-characteristic of fracton systemsleads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization.As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S z ) and its associated dipole moment.First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior.The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S z basis, arising from the interplay of dipole conservation and local interactions.Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states.More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum.We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.